Improving the sparse coding model via hybrid Gaussian priors

Sparse Coding (SC) imposes a sparse prior on the representation coefficients under a dictionary or a sensing matrix. However, the sparse regularization, approximately expressed as the $L_1$-norm, is not strongly convex. The uniqueness of the optimal solution requires the dictionary to be of low mutual coherence. As a specialized form of SC, Convolutional Sparse Coding (CSC) encounters the same issue. Inspired by the Elastic Net, this paper proposes to learn an additional anisotropic Gaussian prior for the sparse codes, thus improving the convexity of the SC problem and enabling the modeling of feature correlation. As a result, the SC problem is modified by the proposed elastic projection. We thereby analyze the effectiveness of the proposed method under the framework of LISTA and demonstrate that this simple technique has the potential to correct bad codes and reduce the error bound, especially in noisy scenarios. Furthermore, we extend this technique to the CSC model for the vision practice of image denoising. Extensive experimental results show that the learned Gaussian prior significantly improves the performance of both the SC and CSC models. Source codes are available at https://github.com/eeejyang/EPCSCNet.